relationship lending and the transmission of monetary...

Relationship Lending and the Transmission of Monetary Policy

Kinda Hachem�y

University of Toronto

First Version September 2008; This Version January 2011

Abstract

Repeated interactions allow lenders to uncover private information about their clients, decreas-

ing the informational asymmetry between a borrower and his lender but introducing one between

the lender and competing �nanciers. This paper constructs a credit-based model of production

to analyze how learning through lending relationships a¤ects the monetary transmission mecha-

nism. I examine how monetary policy changes the incentives of borrowers and lenders to engage

in relationship lending and how these changes then shape the response of aggregate output. A

central �nding is that relationship lending induces a smoother steady state output pro�le and

a less volatile response to certain monetary shocks. This result provides a theoretical basis for

cross-country transmission di¤erences via a relationship lending channel.

Keywords: Relationship Lending, Monetary Transmission, Credit Channels

JEL classi�cation: D82, D83, E37, E44

�Department of Economics, University of Toronto, 150 St. George Street, Toronto, Ontario, Canada,M5S 3G7. E-mail: [email protected].

yThis paper is based on the �rst chapter of my dissertation at the University of Toronto. I thank ShouyongShi for invaluable guidance and supervision. I also thank the associate editor, an anonymous referee, andseminar participants at the University of Toronto, the Federal Reserve Bank of Chicago, and the 2010 SEDAnnual Meeting for helpful suggestions. Financial support from the Social Sciences and Humanities ResearchCouncil of Canada as well as Shouyong Shi�s Bank of Canada Fellowship and Canada Research Chair aregratefully acknowledged. Any remaining errors are my own.

Relationship Lending and the Transmission of Monetary Policy 2

1 Introduction

Macroeconomists have shown how intermediation costs can propagate shocks when lenders

are imperfectly informed about their borrowers. Commonly omitted from the analysis though

is the potential for lenders to learn about their clients through repeated interactions. This

omission is problematic because recent empirical evidence suggests that there is a link be-

tween relationship lending and the transmission of monetary shocks. As de�ned in Boot

(2000), relationship lending is the provision of credit by intermediaries that acquire propri-

etary information about their borrowers over time or across products. Among the major

European economies, Ehrmann et al (2001) establish that relationship lending is much more

prevalent in Germany and Italy than in Spain and France. Incidentally, they also �nd that

the quantity of bank loans responds less severely to a monetary contraction in the �rst two

countries. Borio and Fritz (1995) �nd a similar pattern on the pricing side, with the pass-

through from higher policy rates to higher loan rates occurring more slowly in Germany and

Italy than in Spain.1 A correlation with spending is also visible as Mojon and Peersman

(2003) demonstrate that the peak decline in investment following a monetary contraction is

smaller in Germany and Italy than in Spain and France.

In this paper, I establish a theory of how relationship lending a¤ects the macroeconomic

response to monetary policy. I begin by constructing a credit-based model of production

where lenders can uncover private information about their borrowers� abilities over time.

While such learning decreases the informational asymmetry between a lender and his bor-

rower, it introduces one between the lender and competing �nanciers so the extent and e¤ect

of relationship lending must be endogenously determined. I undertake this determination

and analyze how monetary policy changes the incentives of borrowers and lenders to engage

in such relationships by changing the cost of funds on the interbank market. I then examine

1Additional support for the impact of relationship lending is provided for Italy by Gambacorta (2004)and for Germany by Weth (2002) and Iacoviello and Minetti (2008). Moreover, based on U.S. survey data,Berger and Udell (1995) conclude that American borrowers with larger banking relationships tend to paylower interest rates and are less likely to pledge collateral.


how the response of aggregate output depends on the response of relationship lending. In

contrast to many models of �nancial acceleration, I show that relationship lending prevails in

equilibrium and dampens the canonical credit channel. The prevalence of such relationships

depends not only on the policy rate but also on institutional parameters so, given di¤er-

ences in these parameters, cross-country di¤erences in relationship lending and monetary

transmission are supported.

The analysis uncovers two mechanisms through which relationship lending a¤ects mon-

etary transmission. The �rst mechanism operates during the credit relationship while the

second operates beforehand. As lenders acquire information over the course of their rela-

tionships, they retain only su¢ ciently good borrowers. The presence of other lenders limits

monopoly power and, in order to induce higher repayment rates, it is optimal for informed

lenders to concede positive surplus to some of their borrowers. I demonstrate that this con-

cession includes o¤ering policy-invariant credit terms over intermediate ranges of the policy

rate, giving rise to a �rst mechanism. To be sure, informed lenders do not concede the

entire surplus from relationship lending and, in anticipation of future relationship pro�ts,

unmatched lenders compete more intensely for new borrowers. A second mechanism arises

because this competition lowers loan rates for any given policy, alleviating some of the tight-

ness that information frictions may impart on �rst-time borrowers without actually changing

these frictions in the current period. At an aggregate level, the two mechanisms combine

to produce a smoother steady state output pro�le and a less volatile response to certain

monetary shocks.

The importance of �nancial intermediation for real activity has been emphasized in the

macroeconomics literature.2 However, in analyzing how credit markets transmit shocks

to the real economy, this literature has essentially discounted the propensity of agents in

these markets to engage in relationship lending: Williamson (1987), Bernanke and Gertler

2Perhaps most pointedly, Bernanke (1983) and Diamond and Dybvig (1983) argue that �nancial disruptionpropelled a potentially normal-course recession into the Great Depression. For a survey of the literature onreal-�nancial interactions in business cycles, see Gertler (1988). For a discussion of real-�nancial interactionsin economic development, see King and Levine (1993).


(1989), and Kiyotaki and Moore (1997) abstract from multi-period credit relationships while

Gertler (1992), Khan and Ravikumar (2001), and Smith and Wang (2006) abstract from the

learning bene�ts of such relationships. In contrast, the key feature of multi-period lending

relationships in my model is learning and, in particular, the informational advantage of an

inside lender over all other lenders. I further assume that agents cannot commit ex ante

to long-term contracts, making multi-period lending relationships a sequence of one-period

arrangements whose bene�ts are derived solely from the possibility of lender learning.3 To

the extent that I emphasize relationship lending, this paper is also related to the banking

literature and, in particular, work by Schmeits (2005) and Van Tassel (2002) on the properties

of these relationships. However, neither of these studies investigates how policy rates a¤ect

the resulting contracts or how these contracts then transmit shocks to the macroeconomy,

two questions which are key components of my analysis.4

The rest of the paper proceeds as follows: Section 2 describes the environment in more

detail, Section 3 characterizes the optimal credit decisions, Section 4 determines the resulting

output functions, Sections 5 and 6 discuss the output implications of relationship lending,

and Section 7 concludes. All proofs and derivations are presented in the Appendix.

2 Environment

Time is discrete. All agents are in�nitely-lived, risk neutral, and have discount factor � 2

(0; 1). There is a continuum of �rm types, denoted by ! and distributed over the interval

[0; 1] according to a non-degenerate probability density function f (�). All �rms have access

to the same production technologies: an investment project called P1 and a speculative

3Multiple periods are important here both because they permit learning and because learning has long-term implications. This contrasts with the growth model of Bose and Cothren (1997) where lenders investin learning about borrowers but the information acquired cannot be used in future contracts since agentsare two-period-lived overlapping generations who only enter into credit contracts in their �rst period.

4To this end, I also extend the oft-used two-type banking environment to a continuum of borrower types,permitting non-degenerate lender beliefs and continuous output functions. Moreover, the �rst period creditmarket in my model is competitive, borrowers can choose a di¤erent project each period, and all lenders cancondition their second period loan rates on �rst period default history. While some of the last three featuresare present in either Schmeits (2005) or Van Tassel (2002), neither paper contains all three. Combining theseelements allows me to explore more avenues through which relationship lending can in�uence real activity.


project called P2. Types are private information and high-! �rms are better in the sense

that they are more likely to operate the investment project successfully. In particular, a

type ! who operates P1 is able to produce �1 units of output with probability p (!) and

zero units with probability 1� p (!), where p : [0; 1]! [0; 1] is a continuously di¤erentiable

and strictly increasing function. In contrast, the outcome of the speculative project P2 is

independent of �rm type, yielding �2 with probability q and zero with probability 1 � q.

Assume �2 > �1 and q�2 = p (0) �1 so that the speculative project is riskier in the sense that

it is second order stochastically dominated by the investment project.5 The presence of a

speculative outside option allows the credit contracts described below to a¤ect real activity

by changing the relative attractiveness of safe projects.

To undertake either project, �rms need one unit of capital. Project output is not storable

so this capital must be borrowed from a measure of ex ante identical lenders that also

populates the economy. Lenders cannot operate the production technologies but they have

access to an interbank market for capital. The interest rate in the interbank market is

denoted by r. It is a¤ected by central bank monetary policy and enters the model as the

lenders�cost of funds.6 To simplify the analysis, I assume that r is exogenous and refer to it

directly as the policy rate.

Lenders also have an ability to learn about the borrowers they lend to. In particular, a

lender who has provided credit to a borrower in the past knows more about that borrower

than do all the other lenders. Label the informed or relationship lender an insider and

the other lenders outsiders. I abstract from the process through which insiders acquire

information, summarizing it instead by a positive probability of type discovery. Outsiders

are not privy to the information gathered by insiders. To avoid situations where insiders

design credit contracts to distort the beliefs of outsiders, assume that outsiders are also not

privy to an insider�s o¤er when they make their own o¤ers. They do, however, �nd out if

5p (0) �1 > q�2 requires more algebra but yields similar conclusions.6Essentially, lenders who need more capital can obtain it at the interbank rate while lenders who have

enough capital interpret the rate as an opportunity cost.


a borrower defaulted on a past loan and can revise their beliefs about the borrower�s type

conditional on this information.7

Firms borrow from one lender at a time while lenders can take on more than one borrower.

Agents cannot commit to long-term contracts, inducing a sequence of one-period credit

arrangements. This limits the scope for intertemporal incentives à la Townsend (1982) and

keeps the focus on the learning property of relationship lending.8 Contracts are characterized

by a loan rate so there is no quantity rationing in the model.

There are two notions of time: dates t = 1; 2; ::: denote calendar time in the general

economy while periods k = 1; 2; ::: denote time spent by a borrower in the credit market. As

will be described at the end of this section, borrowers can be exogenously separated from

the credit market so the two notions of time are distinct.

At any date t, �rst-time (i.e., k = 1) borrowers enter the market with private information

about their types. All lenders have the same priors about a new borrower so this borrower

chooses randomly among perfectly competitive lenders. Based on his type and the loan rate

charged, the borrower then chooses which project to undertake. At the end of date t, project

outcomes are realized and debts are settled. Capital is not destroyed in the production

process so the borrowed unit is always recovered by the lender. Interest payments, however,

can only be made by borrowers with successful projects. Lenders cannot observe the exact

output of a project but can detect whether their borrowers have positive consumption so

borrowers settle interest payments if and only if their projects are successful. Default occurs

when a borrower cannot pay interest. Since this happens when his project yields no output,

the probability of default is just the probability of project failure.

Also at the end of date t, a lender learns the type of his �rst-time borrower with proba-

7This is the only cost of default in the model. If the borrower is also forced to wait a few periods beforehis next contract, the marginal type that chooses the risky project may fall but the qualitative conclusionsof the model are unlikely to change.

8Note that the scope is limited but not necessarily eliminated. In the environment presented in thesupplementary appendix, for example, �rst period defaulters end up with higher loan rates the next timearound so there is e¤ectively an intertemporal punishment for choosing the riskier project. However, thepunishment is not complete since uncommitted borrowers can switch to another lender and this new lendermay not �nd it optimal to punish default as much as the original lender would have liked.


bility � 2 (0; 1]. To simplify the exposition, I consider � = 1 here. All other lenders learn

whether or not the borrower defaulted on his �rst period loan. The borrower then moves to

date t + 1 and becomes a second-time (i.e., k = 2) borrower. Conditional on their cost of

funds and their beliefs about the borrower�s type, lenders set their second period loan rates

simultaneously.9 Since insiders and outsiders have di¤erent information sets, the borrower no

longer chooses among perfectly competitive lenders. After receiving all o¤ers, the borrower

decides which contract to accept and which project to then undertake. Once again, project

outcomes are realized and debts are settled.

At the end of date t+ 1, the types of all second-time borrowers are made public. This is

done to avoid carrying credit history throughout the model and, therefore, to keep the state

space �nite. Also starting at the end of the borrower�s second period is a positive probability

� 2 (0; 1) of exogenous separation from the credit market. This separation eliminates all

information about the borrower and requires that he draw a new type and re-enter the market

as a �rst-time borrower in t+2. In contrast, borrowers that are not separated become third-

time borrowers in t + 2. As before, loan rates are determined, lenders and projects chosen,

outcomes realized, and debts settled. Borrowers who do not survive separation at the end

of their third period must start anew while borrowers who do survive it become fourth-time

borrowers in t + 3 and face the same environment they did in t + 2. The market continues

in this way and, even though information is revealed after two periods, the possibility of

exogenous separation ensures that there are always �rst-time, second-time, and advanced

(i.e., k � 3) borrowers at any date t.

The three borrower classes can be interpreted as an approximation of the borrower life

cycle. When a borrower �rst enters the credit market, little is known about him so k = 1

re�ects the market for new borrowers. In contrast, after su¢ ciently many realizations of

the borrower�s credit history, all lenders can form precise beliefs about his type so k � 3

approximates the market for established borrowers. k = 2 captures the intermediate market:

9As we will see in Section 3, the type-independent nature of P2 allows for a pure strategy equilibrium inthe simultaneous game between second period lenders.


after enough time has elapsed for lending relationships to inform insiders but before enough

time has elapsed for credit history to inform outsiders.

3 Optimal Decisions

I now characterize the optimal decisions for each period of the credit market. In what

follows, the value of a lender with cost r and information set about a period k borrower

is denoted by Jk (rj). His optimal loan rate o¤er is then denoted by R�k (rj). It will

be convenient to study the optimization problems recursively, starting with the market for

advanced borrowers.

3.1 Advanced Borrowers (k � 3)

Since the types of advanced borrowers are public, the problem is one of perfect information

for k � 3. Project choice does not a¤ect future outcomes as borrowers either start anew

with exogenous probability � or continue to period k + 1 with exogenous probability 1� �.

Consequently, each borrower will choose the project that yields him a higher expected return

in the current period. A trade-o¤ arises, however, since P1 generates more expected revenue

but also increases the likelihood of interest payments. At high loan rates then, the borrower

may have an incentive to choose the riskier project. Formally, the borrower�s optimal strategy

is characterized relative to a threshold loan rate. For a type ! borrower, the one-period

return to P1 is p (!) [�1 �R] and the one-period return to P2 is q [�2 �R]. The loan rate

that makes him indi¤erent between the two projects is:

R (!) =p (!) �1 � q�2p (!)� q

(1)

where R (0) = 0 and R0(!) > 0. Type ! borrowers thus choose P1 if charged R � R (!)

and P2 otherwise. I summarize this strategy as follows:


(Rj!) =

8><>: p (!) if R � R (!)

q if R > R (!)(2)

Given the borrower�s project strategy, lenders choose the loan rate.10 With probability

�, lenders in k � 3 are separated from their borrowers and must start the next period with

a �rst-timer. All lenders have the same information set in k = 1 so competition forces

expected pro�ts there to zero. Symmetric information in periods k � 3 also means zero

pro�ts. Therefore, the expected revenue of an informed lender who charges his type !

borrower R is (Rj!)R. Proposition 1 establishes the optimal loan rate o¤er:

Proposition 1 De�ne the following critical type:

e! (r) � arg min!2[0;1]

��p (!)R (!)� r�� (3)

In the competition for k � 3 borrowers, lenders o¤er:

R�k�3 (rj!) =

8><>: r=q if ! 2 [0; e! (r))r=p (!) if ! 2 [e! (r) ; 1] (4)

Since p (�)R (�) is monotonically increasing, e! (r) is unique and increasing in r. Moreover,p0 (�) > 0 and p (�) > q imply that lower types are charged higher loan rates at any given

policy rate, consistent with their more costly nature.

3.2 Intermediate Borrowers (k = 2)

In the second period, �rst period credit histories are made public. Denote default by d = D

and non-default by d = N . While this is the only information observed by outside lenders,

an insider also learns his borrower�s type before making an o¤er. This type is revealed to

10Since an advanced borrower operates in an environment of perfect information and homogeneous sepa-ration rates, he attracts the same o¤er from every lender and is thus indi¤erent among them. Without lossof generality, I complete the borrower�s strategy by assuming that he stays with his second period lender forall k � 3.


outsiders at the end of k = 2 though so the borrower�s future outcomes are independent of

current project choice and the optimal strategy of a type ! borrower is still given by (Rj!).

As discussed in Section 2, the game between second period lenders is simultaneous. Con-

sider �rst an insider who has discovered that his borrower is type !. With competition

driving future pro�ts to zero, the insider�s expected revenue is again (Rj!)R. Now, how-

ever, the insider has an informational advantage over all other lenders so competition will not

necessarily eliminate current pro�ts. Noting that the insider can charge above the outsider

rate and lose the borrower, his value is:

J2 (rj!; d) = maxn0; max

R (Rj!)R� r s:t: R � R�2 (rjd)

o(5)

Below R (!), the insider�s revenue is strictly increasing in R. Therefore, if R�2 (rjd) � R (!),

equation (5) reduces to J2 (rj!; d) = max f0; p (!)R�2 (rjd)� rg. Now suppose R�2 (rjd) >

R (!). The insider will either o¤er R = R (!) and induce the selection of P1, o¤er R =

R�2 (rjd) and induce the selection of P2, or o¤er R > R�2 (rjd) and lose the borrower. Without

loss of generality, I assume that insiders only keep borrowers who net them positive expected

pro�t.11 Proposition 2 establishes that the second period credit market splits neatly between

insiders and outsiders. In particular, lending relationships are formed with better borrowers,

consistent with the empirical prediction of Memmel et al (2007) that high quality �rms are

more likely to choose relationship lenders.

Proposition 2 For a given credit history d, outsiders attract second-period borrowers with

! 2 [0; cd (r)] and insiders retain those with ! 2 (cd (r) ; 1], where cd (r) is a threshold

borrower type that satis�es J2 (rjcd (r) ; d) = 0.

Consider now an outsider who has attracted a k = 2 borrower. Represent his beliefs

about the borrower�s type by a cumulative distribution function, bFd (�), de�ned over the11This assumption simpli�es the exposition aimed at here but is innocuous. Assuming instead that the

insider keeps types for which he is indi¤erent yields the same loan rates and output functions derived belowbut the split between borrowers may occur within the insider rather than across insiders and outsiders.


interval [0; cd (r)] according to Bayes�rule. Whatever the borrower�s type, it will be known

to everyone next period so future pro�ts will be competed away. The expected pro�t of an

outsider who charges his group d borrower Rd is thus:R cd(r)0

(Rdjx)Rdd bFd (x)� r (6)

In equilibrium, the outsider�s beliefs will depend on Rd. Moreover, all outsiders must have

the same beliefs so competition also drives (6) down to zero. Proposition 3 summarizes the

outcome of the game between inside and outside lenders vying for second-period borrowers:

Proposition 3 In the competition for k = 2 borrowers, outsiders o¤er R�2 (rjd) = r=q and

get ! 2 [0; e! (r)]. In contrast, insiders keep ! 2 (e! (r) ; 1] by o¤ering:R�2 (rj!; d) =

8><>: R (!) if ! 2 (e! (r) ; b! (r))r=q if ! 2 [b! (r) ; 1] (7)

where e! (r) is as de�ned in Proposition 1 and b! (r) � arg min!2[0;1]

��qR (!)� r��.

Proposition 3 establishes two important results. First, when the insider discovers his bor-

rower�s type with certainty (i.e., � = 1), default history is irrelevant. Second, instead of the

monotonically increasing function of r that would arise in a pooled equilibrium, relationship

lenders charge their borrowers a �at rate over certain ranges of the policy rate.

The adverse selection problem faced by competitive outsiders is key for the �rst result.

Insiders want to keep the best types and, since outsiders cannot observe insider o¤ers before

making their own, no inferences about type can be made based on loan rates. For each

default group then, outsiders know that they will attract the bottom of the distribution so

they o¤er r=q, the maximum competitive rate. This rate is above R (!) for ! < b! (r) so,instead of matching the outsider and inducing P2, the insider can o¤er these borrowers R (!)

and induce P1. Given (3), e! (r) is the lowest type for which undercutting the outsider ispro�table and the market splits according to it.12

12Note, however, that d would not be irrelevant if � < 1. Instead, � < 1 implies a positive probability thatno one is informed about the borrower�s type in k = 2, making credit history the only piece of informationavailable to the market and mitigating the adverse selection problem.


The second result from Proposition 3 is illustrated in Figure 1(a). Details on the con-

struction of the �gure are provided in the Appendix. Consider type !A as shown on the

vertical axis. If r > rB, then !A falls below e! (r) and the borrower moves to an outsider.If r � rB, then !A stays with his insider and, for policy rates between rA and rB, he is

charged his reservation loan rate. For policy rates below rA, the insider charges him r=q

but !A is su¢ ciently high that this rate is less than R (!A). Notice the role of relationship

lending here: by revealing type to the insider, it allows him to gauge how much can be

extracted from the borrower without inducing the risky project. In equilibrium, policy rates

between rA and rB are such that the insider pro�ts from using information generated by his

lending relationship with !A to undercut the outsiders. The shaded region in Figure 1(a)

illustrates that the length of the interval over which a borrower is charged his reservation

rate is increasing in the borrower�s type. Figure 1(a) also demonstrates that the proportion

of types charged their reservation rate as a result of relationship lending exhibits a hump-

shaped response to increases in the policy rate (i.e., the vertical distance between e! (r) andb! (r) rises then falls). As r increases, the marginal type on which an insider breaks evenrises so the fraction of borrowers admitted into lending relationships falls. Within the group

of relationship borrowers, however, the insider wants to increase the marginal type that he

undercuts on. Initially, the second e¤ect dominates the �rst and the vertical distance rises

but, eventually, the �rst e¤ect dominates the second and the vertical distance falls.

Before proceeding to k = 1, let us elaborate on the role of outsiders in these results. The

free entry of other lenders forces the insider to solve a constrained optimization problem

and, with monopoly rents precluded, insiders choose to tailor contracts around reservation

rates in the manner discussed above.13 The absence of monopoly rents, however, does not

mean the absence of all rents: (3) and q < p (!) imply R�2 (rj!; d) > r=p (!), where r=p (!)

13In other words, competition prevents the borrower from being informationally captured and, as inSchmeits (2005), mitigates the hold-up problem. The result that competition can help sustain a mutuallybene�cial second-period credit contract contrasts somewhat with the Petersen and Rajan (1995) argumentthat concentration increases the value of lending relationships. Therefore, consistent with Cao and Shi (2001),the treatment of information appears critical in analyzing interactions between credit market structure andcredit market outcomes.


is the zero-pro�t loan rate derived in Subsection 3.1 for ! > e! (r). Moreover, the fact thatthe insider simply matches the outsider rate for types above b! (r) is consistent with theempirical �nding of Bharath et al (2009) that prices of relationship and non-relationship

loans are indistinguishable for borrowers at the top of the asset size distribution.14 Here

though, b!0 (r) > 0 so the fraction of borrowers for which insider and outsider prices are

indistinguishable declines with the policy rate.

3.3 New Borrowers (k = 1)

Recall from Proposition 3 that the second period equilibrium does not depend on default

history when � = 1. Therefore, the reservation rate of a �rst-time borrower is still R (!)

and his project strategy is once again (Rj!) as de�ned in (2). Moreover, the d�s drop out

of (5) and the insider�s valuation of a second period contract with a type ! borrower can be

denoted by J2 (rj!). Assuming for now that the future policy rate is expected to be the same

as the current one, �rst period lenders obtain the following expected pro�t from charging

their borrowers R1:

R 10 (R1jx)R1dF (x) + �

R 1e!(r)J2 (rjx) dF (x)� r (8)

Let R1 (r) denote the equilibrium �rst period loan rate and de�ne type � (r) such that

R (� (r)) = R1 (r).15 By this de�nition, all types above � (r) choose P1 and all types below

it choose P2. Since competition between identically uninformed lenders drives (8) down to

zero, � (r) is characterized by:

� (r) � arg min!2[0;1]

��hR !0 qdF (x) + R 1! p (x) dF (x)iR (!) + �R 1e!(r)J2 (rjx) dF (x)� r

�� (9)

where (5) and Proposition 3 can be used to substitute out J2 (rjx). Further discussion of

� (r) is deferred until Section 4. For now though, note that a continuous function over a

14To the extent that ability and assets are positively correlated, the distribution of assets can be viewedas one approximation of the distribution of types.15If R1 (r) > R (1), then � (r) is corner at 1.


compact set has at least one argmin and, as demonstrated in the Appendix, the argmin

that de�nes � (r) is unique and non-decreasing under the following regulatory conditions:

Assumption 1 f (�) is well-behaved with f (1) � 1

Assumption 2 p (0) � q + [p(1)�q]3

qp0(1)+[p(1)�q]2

Assumption 3 p00 (�) is su¢ ciently low

Assumption 4 p (1) � 2q

Assumption 1 precludes the economy from having a disproportionately large group of

high types. It is a relatively innocuous assumption, satis�ed by both uniform and truncated

normal distributions over the unit interval. In Section 2, we imposed p (0) > q so Assumption

2 just restricts the margin by which p (0) exceeds q. Assumption 3 says that p (�) is either

concave, linear, or mildly convex. In other words, while the probability of succeeding in

the investment project increases with �rm type, it does not increase exponentially. Finally,

Assumption 4 regulates the outside option by putting a lower bound on q.

4 Output Functions

From the preceding analysis, there are two channels through which relationship lending

can a¤ect economic activity. First and as demonstrated in Subsection 3.2, insiders use

the information they acquire over the course of their relationships to tailor loan rates non-

monotonically. Second and as visible in Subsection 3.3, the loan rate for new borrowers

depends on expectations of future relationship pro�ts. Having determined the e¤ect of rela-

tionship lending on credit terms, let us now formalize its e¤ect on the output produced under

these terms. As will become apparent below, the key variables for the output calculation are

the cuto¤ types e! (r) and � (r).


4.1 Advanced Borrowers (k � 3)

Proposition 1 established that R�k�3 (rj!) = r=p (!) for all ! � e! (r) and, according to theborrower strategy in (2), this loan rate will induce investment in P1 as long as r=p (!) �

R (!). Given (3), the latter condition is guaranteed by ! � e! (r) so we can conclude thatall advanced borrowers above e! (r) choose P1. Consider now ! < e! (r). Proposition 1 alsoestablished that R�k�3 (rj!) = r=q for borrowers below e! (r) so they will choose P2 as long asr=q > R (!). A su¢ cient condition for this inequality is r > p (!)R (!), which is guaranteed

by ! < e! (r). Normalized by population then, the total output of advanced borrowers is:Yk�3 (r) =

R e!(r)0

q�2dF (x) +R 1e!(r)p (x) �1dF (x) (10)

With e!0 (r) > 0 and p (x) �1 > q�2 for x 2 (0; 1], equation (10) de�nes an output function

that is decreasing in the policy rate.

4.2 Intermediate Borrowers (k = 2)

Proposition 3 established that intermediate borrowers below e! (r) are �nanced by outsidersat loan rate r=q. From the previous subsection, we know that r=q induces types below

e! (r) to choose the speculative project so we can conclude that all intermediate borrowersbelow e! (r) undertake P2. Proposition 3 also established that intermediate borrowers with! 2 (e! (r) ; b! (r)) are �nanced by insiders at loan rate R (!) so, given (2), they clearly chooseP1. Consider now intermediate borrowers above b! (r). These types are �nanced by insidersat loan rate r=q so they will also choose P1 as long as r=q < R (!). With b! (r) as de�ned inProposition 3, the latter condition is guaranteed by ! � b! (r) and we can conclude that allintermediate borrowers above e! (r) choose P1. In other words, Y2 (r) is also given by (10).4.3 New Borrowers (k = 1)

As discussed in Subsection 3.3, the loan rate for �rst-time borrowers induces all types above

� (r) to choose P1 and all types below it to choose P2. Therefore, the total output of new


borrowers is:

Y1 (r) =R �(r)0

q�2dF (x) +R 1�(r)p (x) �1dF (x) (11)

With �0 (r) > 0, equation (11) also de�nes an output function that is decreasing in the policy

rate.

4.4 Aggregation

There are new, intermediate, and advanced borrowers at any date t so we must determine

the distribution of borrowers across periods in order to calculate aggregate output. Without

loss of generality, set the population size to one and let 1;t, 2;t, and k�3;t denote the

proportions of period 1, 2, and k � 3 borrowers at date t. Aggregate output can then be

written as:

Y (r; t) = 1;tY1 (r) + 2;tY2 (r) + k�3;tYk�3 (r)

where Y1 (r) and Y2 (r) = Yk�3 (r) are given by (11) and (10) respectively. With the pos-

sibility of exogenous separation beginning at the end of the second period, the distribution

evolves according to:

1;t+1 = � (2;t +k�3;t)

2;t+1 = 1;t

k�3;t+1 = 1�1;t+1 �2;t+1

Substituting k�3;t into the expression for 1;t+1, the evolution of 1 is determined by a

one-dimensional di¤erence equation and, with � 2 (0; 1), the entire system is asymptotically

stable. Therefore, starting from any initial distribution, the proportions converge to 1 =

2 =�1+�

and k�3 =1��1+�. Steady state aggregate output is thus:

Y (r) =�

1 + �Y1 (r) +

1

1 + �Yk�3 (r) (12)


Moreover, the extent of relationship lending is captured by (1� e! (r)) = (1 + �) and isdecreasing in the policy rate and the rate of exogenous separation.

4.5 Benchmark for Comparison

To better appreciate the macroeconomic e¤ects of relationship lending, it will be instructive

to compare the results of my model to those of a standard credit channel model where

exogenous separation occurs with certainty every period and private information is never

revealed. In this context, a representative lender�s expected pro�t from charging RS isR 10 (RSjx)RSdF (x) � r. Competition drives this expression down to zero and yields an

equilibrium loan rate denoted by RS (r). De�ning type � (r) such that R (� (r)) = RS (r),

the solution to the standard model and the resulting output function are characterized by

(13) and (14) respectively:

� (r) � arg min!2[0;1]

��hR !0 qdF (x) + R 1! p (x) dF (x)iR (!)� r�� (13)

YS (r) =R �(r)0

q�2dF (x) +R 1�(r)

p (x) �1dF (x) (14)

5 Output Implications by Borrower Class

Given the output functions derived above, the e¤ect of relationship lending on per-period

output can be gauged by comparing e! (r), � (r), and � (r). The key properties of these cuto¤sare derived in the Appendix and illustrated in Figure 1(b). The implied output functions

are then shown in Figure 2(a).16

5.1 Output of Advanced and Intermediate Borrowers

Begin with e! (r) and � (r). Aside from the corners, � (r) only intersects e! (r) once. Moreover,� (r) approaches this intersection from below e! (r). Given the output functions in (10) and16Since all output functions were normalized by population, they are directly comparable.


(14), we can then conclude that YS (r) is greater than Yk�3 (r) for low policy rates but less

than Yk�3 (r) otherwise. The di¤erence between Yk�3 (r) and YS (r) is intuitive. At low

policy rates, informed lenders can grant favourable credit terms (i.e., loan rates low enough

to induce P1 ) to more types without su¤ering a loss. The same is true for uninformed

lenders in the standard model but, since they can only o¤er a pooled rate, some of the lower

types who would not otherwise receive favourable terms now do. In contrast, when the cost

of funds is su¢ ciently high, this mechanism has the opposite e¤ect, yielding � (r) > e! (r)and YS (r) < Yk�3 (r). As can be gleaned from Subsection 3.2, relationship lending results in

credit terms which push Y2 (r) towards Yk�3 (r) so the �rst macroeconomic impact of these

relationships is a less severe second period output pro�le relative to the standard model.

5.2 Output of New Borrowers

Consider now � (r) and � (r). Aside from the corners, � (r) does not intersect � (r). Instead,

� (r) < � (r) which implies Y1 (r) > YS (r). Therefore, even though the �rst period of the

baseline model is characterized by the same information frictions as the standard one, it

generates higher output. To see why, note that �rst period lenders compete more �ercely for

borrowers in anticipation of the second period insider pro�ts a¤orded by relationship lending.

As a result, the pooled rate is driven down further, a greater number of types opt for P1, and

the second macroeconomic impact of relationship lending is an improvement in �rst period

output relative to the standard model. This e¤ect is most pronounced over moderate policy

rates since very high values of r are associated with few lending relationships while very low

values of r provide only limited scope for further reductions in the pooled rate.

6 Implications for Aggregate Output

How do these e¤ects roll up into economy-wide output? In this section, I establish that

relationship lending leads to a smoother steady state aggregate output function and a less

dramatic response to certain monetary shocks.


6.1 Steady State

Aggregate steady state results are illustrated in Figure 2(b). With a higher probability of

exogenous separation, there is a greater mass of �rst-time borrowers and the bene�ts of

relationship lending are more visible at moderate policy rates. In contrast, lower separation

probabilities push Y (r) towards Yk�3 (r) and magnify the bene�ts of relationship lending at

higher policy rates. For any value of � though, Y (r) is better than YS (r) in two respects.

First, Y (r) is smoother over the interval r 2�0; qR (1)

�and, second, Y (r) > q�2 = YS (r)

over the interval r 2�qR (1) ; p (1)R (1)

�. Proposition 4 establishes the smoothness result

more formally. To simplify the exposition both here and in the next subsection, I strengthen

Assumptions 1 and 3 so that f (�) = 1 and p00 (�) = 0.

Proposition 4 Let smoothness be a notion of curvature and de�ne the smoothness of y (r)

over r 2 [a; b] by S �R bajy00 (r)j dr. It can be shown that

R qR(1)0

jY 00 (r)j dr <R qR(1)0

jY 00S (r)j dr,

implying that Y (r) is smoother than YS (r). The same conclusion holds if smoothness is taken

to be a notion of dependence and de�ned by bS � R bajy0 (r)j dr.

In words, Proposition 4 says that relationship lending smooths the aggregate output

function both by decreasing the average curvature of this function and by decreasing the

average dependence of steady state output on policy rates.

To see that the smoothing e¤ect does not result from the timing of exogenous separation

or the fact that information is eventually revealed to all lenders, suppose that separation

occurs with probability � at the end of the �rst period and probability 1 at the end of

the second. The k = 2 problem is unchanged so Y2 (r) is still given by (10). The k = 1

problem is slightly di¤erent since the second term in (8) must now be multiplied by (1� �).

Once again though, higher values of � imply that the economy is able to sustain fewer

lending relationships so weight shifts from Y2 (r) = Yk�3 (r) to Y1 (r) and the aggregate

output function steepens. Moreover, with exogenous separation beginning at the end of

period 1, higher values of � lead �rst period lenders to view future relationship pro�ts as less


likely, pushing � (r) closer to � (r), Y1 (r) closer to YS (r), and actually hastening the fall in

output. Therefore, the institutional parameters that a¤ect an economy�s ability to sustain

lending relationships a¤ect its real response to policy. Here, � can be interpreted as the �rm

death rate and, as suggested by Adachi and Aidis (2007), its magnitude is in�uenced by

the regulatory environment (i.e., enforcement of property rights, anti-trust laws, hiring and

�ring restrictions, predatory tax practices, inspection agencies, etc.).17

6.2 Dynamics

Let us now investigate whether relationship lending also fosters smoothness in a stochastic

environment. In particular, if the policy rate follows an AR(1) process, does the presence of

relationship lending generate a less volatile output response to temporary shocks? As proven

in the Appendix and summarized in the following proposition, the answer is unambiguously

yes for small shocks in an already contractionary environment.18 That is, when liquidity is

low, relationship lending does indeed dampen the transmission of monetary policy and, for

that matter, the transmission of further liquidity shocks.

Proposition 5 Suppose the policy rate evolves according to rt = rss + � (rss � rt�1) + "t,

where rss is its steady state value, � captures the speed of mean reversion, and "t is an IID

shock. If rss is high, then a small temporary shock �that is, a one-time shock that keeps the

policy rate between 0 and qR (1) �induces a less volatile transition path when relationship

lending is present.

Consider now a permanent shock. As long as the policy rate is expected to stay at its

post-shock level, the total output of each borrower class �new, intermediate, and advanced

�adjusts immediately to its new steady state. Recalling the laws of motion presented in

17In a more general version of the model, both � and � would qualify as institutional parameters with �representing average lender quality.18The e¤ects in an expansionary environment are not as clear. For example, if rss is very low and the

shock is small enough, then the output of new borrowers will exhibit less volatility while that of intermediateand advanced borrowers will exhibit more.


Subsection 4.4, the policy rate does not a¤ect the distribution of borrowers across these

classes so aggregate output also adjusts immediately after a permanent shock. In light of

Proposition 4, the size of the adjustment will often be smaller in the presence of relationship

lending so there is still a sense in which the transition is less pronounced.

This, however, raises the question of how distributional dynamics would a¤ect the aggre-

gate output response when credit markets feature relationship lending. To gain some insight

into this issue, I consider the transmission of a permanent shock when the current model

is extended to allow for di¤erent separation rates. In particular, I assume that borrowers

who stay with their insiders have a lower probability of exogenous separation. The results so

far suggest that this assumption is not unreasonable. As demonstrated above, relationship

lending makes loans more accessible to new borrowers and sometimes induces lenders with

intermediate borrowers not to increase loan rates. Since better credit terms may help bor-

rowers overcome idiosyncratic events that would have otherwise put them out of business,

relationship lending may indeed foster lower �rm exit rates.

The di¤erence in separation rates has two important implications. First and as noted

above, it allows us to consider distributional dynamics. We have already seen that the policy

rate a¤ects which borrowers enter into lending relationships so, when separation rates di¤er

between insiders and outsiders, the policy rate will also a¤ect the distribution of borrowers

across periods. Second, having di¤erent separation rates gives insiders more bargaining power

over high types. A separated borrower must draw a new type and re-enter the credit market

as a �rst-timer so separation is very costly for high-! �rms. By supporting a lower exit rate,

insiders can now charge slightly above the outsider o¤er without losing these borrowers. The

details of the extended model are provided in a supplementary appendix. Although the

introduction of more bargaining power complicates the analysis, numerical results suggest

that relationship lending induces distributional dynamics which then foster a more gradual

output response to certain permanent shocks.


7 Conclusion

This paper has constructed a credit-based model of production to examine how relationship

lending a¤ects the monetary transmission mechanism. I analyzed how monetary policy

changes the incentives of borrowers and lenders to engage in lending relationships and how

these changes then shape the response of aggregate output. I �nd that su¢ ciently good

borrowers enter into lending relationships and, over intermediate ranges of the policy rate,

their loan rates are policy-invariant and preferable to the terms o¤ered by uninformed lenders.

In addition, competition among lenders for future relationship pro�ts alleviates some of the

tightness that could otherwise arise in the market for new borrowers. On average then, the

informational properties of relationship lending lead to improved credit terms and economies

that can sustain these relationships have a smoother steady state aggregate output pro�le

and a less dramatic response to certain monetary shocks. These results provide a theoretical

basis for cross-country transmission di¤erences via a relationship lending channel so future

work will be directed at calibrations to quantify the e¤ect.


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Figure 1: Critical Types

(a) Graphical representation of Proposition 3

(b) Comparison of critical types


Figure 2: Steady State Output

(a) Per-period output functions

(b) Aggregate output functions


Appendix: Proofs and Derivations

This appendix proves Propositions 1 to 5 and derives the key features of Figure 1. The

material is presented in the order it is referenced in the main text.

A. Proof of Proposition 1

The highest sustainable loan rate is �2: anything above �2 and the borrower will not

want to undertake either project. For loan rates less than or equal to �2, the lender�s

revenue function is:

(Rj!)R =

8><>: p (!)R if R � R (!)

qR if R > R (!)

In equilibrium, competition forces (Rj!)R = r so the zero-pro�t loan rates are:

R =

8><>: r=p (!) if r � p (!)R (!)

r=q if r > qR (!)

In other words, the lender charges r=p (!) if r 2�0; qR (!)

�, either r=p (!) or r=q if

r 2�qR (!) ; p (!)R (!)

�, and r=q if r 2

�p (!)R (!) ; �2

�. I now show that there exists

a pro�table deviation from r=q when r 2�qR (!) ; p (!)R (!)

�, ruling it out as an equilib-

rium o¤er. In particular, suppose the lender o¤ers R (!) instead of r=q. Since r > qR (!), the

lender can indeed do this without losing the borrower. Moreover, at R (!), type ! chooses

P1 so the lender�s expected pro�t is p (!)R (!)� r > 0, con�rming the pro�table deviation.

Therefore, the equilibrium loan rate for k � 3 is:

R�k�3 (rj!) =

8><>: r=p (!) if r � p (!)R (!)

r=q if r > p (!)R (!)

The monotonicity of p (�)R (�) and the de�nition of e! (r) then yield the form in (4). �


B. Proof of Proposition 2

Suppose types !0 and !0 + � have the same credit history d. Since credit history is

the only information that the outsider can condition on, he o¤ers !0 and !0 + � the same

loan rate R�2 (rjd). Consider now an insider who �nds it optimal to keep !0. Retaining !0

when exogenous separation rates are homogeneous implies that the insider must be charging

R�2 (rj!0; d) � R�2 (rjd). Moreover, the fact that the insider �nds it optimal to keep !0 means

that he must be making positive pro�t on this borrower. With � > 0 and R0(!) > 0,

equation (2) establishes that (Rj!0 + �) � (Rj!0) for any R. Therefore, the insider could

o¤er !0 + � loan rate R�2 (rj!0; d), keep him, and make at least as much as he is making on

!0. Since !0 and � were chosen arbitrarily, Proposition 2 follows by induction. �

C. Proof of Proposition 3

Let Rd (r) denote the as yet undetermined solution to the outsider�s problem and de�ne

type !d (r) such that R (!d (r)) = Rd (r). By de�nition, all types above !d (r) choose P1

and all types below it choose P2 so setting (6) to zero yields:

Rd (r) =

8><>: r=hR !d(r)0

qd bFd (x) + R cd(r)!d(r)p (x) d bFd (x)i if !d (r) � cd (r)

r=q if !d (r) > cd (r)(C.1)

The proof of Proposition 3 proceeds in three steps. First, I prove R�2 (rjd) = r=q. Second, I

prove cd (r) = e! (r). Third, I establish equation (7). The following Lemma will be useful forthe �rst step:

Lemma 1 bFd (�) is the CDF of a non-degenerate distribution.Proof. Using Bayes�Rule, bFd (x) is given by:

bFd (x) � Pr (! � xjd) = Pr (dj! � x) Pr (! � x)

Pr (d)


All �rst-time borrowers advance to the second period so the unconditional type distribution

is F (�). Note, however, that Pr (! � x) = F (x) =F (cd (r)) since the outsider only gets

! 2 [0; cd (r)]. If cd (r) � � (r), where � (r) is the lowest type that chose P1 in the �rst

period, then the outsider�s beliefs are given by:

bFd (x) = F (x)

F (cd (r))for x 2 [0; cd (r)]

On the other hand, if cd (r) > � (r), then Bayesian updating yields:

bFN (x) =8>>>><>>>>:

qF (x)

qF (� (r)) +R cN (r)�(r)

p (z) dF (z)if x 2 [0; � (r))

qF (� (r)) +R x�(r)

p (z) dF (z)

qF (� (r)) +R cN (r)�(r)

p (z) dF (z)if x 2 [� (r) ; cN (r)]

bFD (x) =8>>>><>>>>:

(1� q)F (x)

(1� q)F (� (r)) +R cD(r)�(r)

(1� p (z)) dF (z)if x 2 [0; � (r))

(1� q)F (� (r)) +R x�(r)(1� p (z)) dF (z)

(1� q)F (� (r)) +R cD(r)�(r)

(1� p (z)) dF (z)if x 2 [� (r) ; cD (r)]

With F (�) well-behaved, bFd (�) is the CDF of a non-degenerate distribution. �We can now establishR�2 (rjd) = r=q. Given (C.1), it will be enough to prove that !d (r) >

cd (r). The proof proceeds by contradiction. In particular, if !d (r) � cd (r), then a type

cd (r) borrower will choose P1 if o¤ered loan rate Rd (r), permitting his insider an expected

pro�t of � = p (cd (r))Rd (r)� r. From Lemma 1, we know that bFd (�) is not degenerate atcd (r) so !d (r) � cd (r) in (C.1) implies Rd (r) > r=p (cd (r)) and, therefore, � > 0. From

Proposition 2 though, J2 (rjcd (r) ; d) = 0 so the fact that J2 (rj�) is a maximum value function

implies � � 0. By contradiction then, !d (r) > cd (r) and R (cd (r)) < R�2 (rjd) = r=q.

To determine the value of cd (r), note that an insider with a type cd (r) borrower can

make an expected pro�t of p (cd (r))R (cd (r)) � r � J2 (rjcd (r) ; d) = 0 by charging him


R (cd (r)). If the inequality is strict, then there exists a � > 0 such that R (cd (r) + �) < r=q

and p (cd (r) + �)R (cd (r) + �) � r < 0. Therefore, the best an insider can do with a type

cd (r) + � borrower is charge R�2 (rjd) = r=q and break even but, given that insiders only

keep borrowers which yield them positive pro�t, this contradicts the fact that they keep all

! > cd (r). In equilibrium then, cD (r) and cN (r) are implicitly de�ned by p (�)R (�) = r.

This is the same equation that de�nes e! (r), implying cD (r) = cN (r) = e! (r).Since we now know that the insider only keeps ! > e! (r), we can restrict attention to

r < p (!)R (!) in order to prove (7). Consider �rst r 2�qR (!) ; p (!)R (!)

�. From the

proof of Proposition 1, we know that an informed lender would rather charge R (!) than

r=q for these policy rates. Therefore, since the insider cannot charge above the outsider

o¤er of R�2 (rjd) = r=q and keep the borrower, he will charge R�2 (rj!; d) = R (!) and get

J2 (rj!; d) = p (!)R (!) � r > 0. Consider now r � qR (!). In this case, the outsider�s

o¤er falls below the borrower�s reservation loan rate and the best the insider can do is

match it, yielding R�2 (rj!; d) = r=q and J2 (rj!; d) = (p (!)� q) r=q. Recalling that r 2�qR (!) ; p (!)R (!)

�corresponds to ! 2 (e! (r) ; b! (r)) and r � qR (!) corresponds to ! �

b! (r) produces R�2 (rj!; d) as in (7). �D. Construction of Figure 1

(a) Consider �rst b! (r). At r = 0, we need qR (b! (0)) = 0 which occurs if and only if

b! (0) = 0. Since R (�) is increasing, b! (r) is increasing for r 2 �0; qR (1)� and equal to 1for r � qR (1). Consider now e! (r). Once again, r = 0 yields e! (0) = 0. The fact that

p (�)R (�) is increasing implies that e! (r) is increasing for r 2 �0; p (1)R (1)� and equal to 1for r � p (1)R (1). At policy rate r, the set of types for which an insider charges reservation

rates has length jb! (r)� e! (r)j (i.e., the vertical distance between the b! (r) and e! (r) curves).Since ! 2 (e! (r) ; b! (r)) if and only if r 2 �qR (!) ; p (!)R (!)�, the range of policy rates overwhich an insider charges type ! his reservation rate has length

��p (!)R (!)� qR (!)�� (i.e.,

the horizontal distance between the b! (r) and e! (r) curves). Having p (�) > q establishes


e! (r) � b! (r) and having p0 (�) > 0 establishes dd!

��p (!)R (!)� qR (!)�� > 0.

(b) Consider � (r). As before, � (0) = 0 is immediate. De�ne:

g (!) �hR !0qdF (x) +

R 1!p (x) dF (x)

iR (!)

Note that q�2 = p (0) �1 allows us to write:

g0 (!) = � [p (!)� p (0)] f (!) �1 +

�p0 (!) [p (0)� q]

[p (!)� q] [p (!)� p (0)]

�g (!)

Since g (0) = 0 and g (!) > 0 for all ! 2 (0; 1], it must be the case that g0 (0) > 0. Lemma 2

establishes the sign of g0 (!) over ! 2 (0; 1) under the assumptions in Subsection 3.3:

Lemma 2 Under Assumptions 1 to 3, g (!) is monotonically increasing over ! 2 (0; 1).

Proof. If the claim is not true, then there must be at least one z 2 (0; 1) such that

g0 (z) = 0. Using the expression for g0 (�) presented above, this z is implicitly de�ned by

g (z) = [p(z)�q][p(z)�p(0)]2f(z)�1p0(z)[p(0)�q] . Denoting the right side of this equation by h (�), we can say

that g (�) achieves a local optimum every time it intersects h (�). In other words, g0 (�)

switches signs at any z such that g (z) = h (z). Recalling g0 (0) > 0, it follows that g (�)

begins to decrease after its �rst intersection with h (�). Under the assumptions, h0 (�) > 0

and there can be no further intersections. Therefore, if there is indeed a z 2 (0; 1) such

that g (z) = h (z), then we will have g0 (1) < 0. Taken together, g0 (1) < 0 and f (1) � 1

imply p (0) < q + [p(1)�q]3

qp0(1)+[p(1)�q]2 , violating Assumption 2. As a result, there cannot be a

z 2 (0; 1) such that g (z) = h (z) and, by implication, g (�) must be monotonically increasing

over ! 2 (0; 1). �

With g (!) increasing, the minimization problem in (13) yields only one argmin. More-

over, � (r) is increasing for r 2�0; qR (1)

�and equal to 1 for r � qR (1). Consider now

all points such that e! (r) = � (r) = z. Using the de�nitions of e! (r) and � (r), any suchz 2 (0; 1) must satisfy p (z) =

hR z0qdF (!) +

R 1zp (!) dF (!)

i� u (z). Since u0 (�) < 0,


p0 (�) > 0, u (0) > p (0), and u (1) < p (1), we know that p (�) and u (�) intersect only once.

Therefore, there is only one point such that � (r) = e! (r) 2 (0; 1). Since both � (r) and e! (r)are increasing, this implies that there is only one r 2

�0; p (1)R (1)

�such that � (r) = e! (r).

Moreover, � (0) = e! (0) = 0 and e!0 (0) = 1

p(0)R0(0)

> 1

u(0)R0(0)

= �0 (0) imply that � (r)

approaches its intersection with e! (r) from below e! (r).Consider now � (r). We can rewrite (9) as:

� (r) = arg min!2[0;1]

��g (!)� hr � �R 1e!(r)J2 (rjx) dF (x)i��

By Lemma 2, g0 (!) > 0 so this argmin is unique and � (0) = 0. Furthermore, � (r) de�ned

as the argmin of jg (�)� rj and g (1) = qR (1) imply � (r) < � (r) for all r 2�0; qR (1)

�.

At r = p (1)R (1) though, e! (r) = 1 so � (r) = � (r) = 1. Therefore, � (r) reaches 1 for

r 2�qR (1) ; p (1)R (1)

�. Now, using Proposition 3:

R 1e!(r)J2 (rjx) dF (x) = R b!(r)e!(r) �p (x)R (x)� r�dF (x) +

R 1b!(r) hp(x)q � 1irdF (x)

Assumption 4 ensures that r��R 1e!(r)J2 (rjx) dF (x) is increasing in r so �(r) is also increasing

until it reaches 1. Let us now examine all policy rates such that � (r) = e! (r) 2 (0; 1). Usingthe de�nitions of e! (r) and � (r), any such r must satisfy:

1p(e!(r))

hR e!(r)0

qdF (x) +R 1e!(r)p (x) dF (x)i+ �

R 1e!(r) J2(rjx)rdF (x) = 1

The �rst term on the left hand side is clearly decreasing in r. The second term is also

decreasing since ddr

R 1e!(r) J2(rjx)rdF (x) = �

R b!(r)e!(r) p(x)R(x)r2dF (x) < 0. Therefore, the left hand

side (LHS) is decreasing in r while the right hand side (RHS) is constant. Moreover, at

r = 0, we have LHS = 1 > 1 = RHS and, at r = p (1)R (1), we have LHS = qp(1)

< 1 =

RHS. Therefore, there is a unique r 2�0; p (1)R (1)

�such that e! (r) = � (r). Moreover,

since � (r) � � (r) and � (r) is initially below e! (r), we know that � (r) also approaches itsintersection with e! (r) from below e! (r).


Note that Assumption 3 also identi�es the curvature of e! (r). Consider r 2 �0; p (1)R (1)�.Di¤erentiating both sides of p (e!)R (e!) = r yields de!

dr= p(e!)�q

p0(e!)[2p(e!)�1�q�2�r] and, therefore,d2e!dr2/ 2p0 (e!) q h �1�R(e!)

2p(e!)�1�q�2�ri� p00(e!)

p0(e!) [p (e!)� q]. It can be shown that R (e!) < �1 so the �rst

term in d2e!dr2

is positive and, with p00 (�) su¢ ciently low, e! (r) is convex. A similar procedureestablishes the convexity of b! (r) over r 2 �0; qR (1)�. All other properties of e! (r) and b! (r)carry over from part (a).

E. Proof of Proposition 4

I start by establishing the convexity of � (r) and � (r) over policy rates where these

cuto¤s are interior (recall that the convexity of e! (r) was established in Section D). With� (r) de�ned by

hq� +

R 1�p (x) dx

iR (�) = r, we have:

d�

dr=

1

� [p (�)� p (0)] �1 +hq� +

R 1�p (x) dx

iR0(�)

d2�

dr2= �0 (r)3

�p0 (�) �1 + [p (�)� q]R

0(�)�

hq� +

R 1�p (x) dx

iR00(�)�

From Section D, �0 (r) > 0. Moreover, R00(!) / p00 (!) [p (!)� q] � 2p0 (!)2 so Assumption

3 ensures R00(!) < 0 and, thus, �00 (r) > 0. Turn now to � (r). Di¤erentiating the equation

which implicitly de�nes � (r) yields:

d�

dr=

1� �A0 (r)

� [p (�)� q]R (�) +hq� +

R 1�p (x) dx

iR0(�)

where: A (r) �R b!(r)e!(r) �p (x)R (x)� r

�dx+

R 1b!(r) hp(x)q � 1irdx

After some algebra, we get that the second derivative satis�es:

d2�

dr2/ ��A00 (r) +

hp0 (�)R (�) + 2 [p (�)� q]R

0(�)�

hq� +

R 1�p (x) dx

iR00(�)i�0 (r)2


Since R00(�) < 0, a su¢ cient condition for �00 (r) > 0 is A00 (r) < 0. With p (�) > q and

A00 (r) = e!0 (r)� p(b!(r))qb!0 (r), this su¢ cient condition is guaranteed by e!0 (r) < b!0 (r). Note

that e!0 (r) < b!0 (r) is equivalent to p0 (e!)R (e!) + p (e!)R0 (e!) > qR0(b!), which is certainly

true since e! � b! and R00 (�) < 0 imply that R0 (e!) > R0(b!). Thus, A00 (r) < 0 and �00 (r) > 0.

Turn now to the relevance of these results for steady state smoothness. That � (r) is an

increasing and convex function of r yields:

Y 0S (r) = � [p (1)� p (0)] �1� (r) �

0 (r) < 0

Y 00S (r) = � [p (1)� p (0)] �1

��0 (r)2 + � (r) �00 (r)

�< 0

Moreover, since � (r) and e! (r) are also increasing and convex, we have:Y 0 (r) = � [p (1)� p (0)] �1

h�1+�

� (r) �0 (r) + 11+�e! (r) e!0 (r)i < 0

Y 00 (r) = � [p (1)� p (0)] �1

h�1+�

��0 (r)2 + � (r) �00 (r)

�+ 1

1+�

�e!0 (r)2 + e! (r) e!00 (r)�i < 0The concavity of YS (r) and Y (r) simpli�es the smoothness measure. In particular, if y (r)

is a concave function, we can write:

S = �Z b

a

y00 (r) dr = � [y0 (b)� y0 (a)]

The right-hand side limits of Y 0 (0) and Y 0S (0) are both zero so establishing the desired

smoothness result amounts to establishing Y 0 �qR (1)� > Y 0S

�qR (1)

�. Consider �rst � = 0.

In this case, we want to establish e! �qR (1)� e!0 �qR (1)� < ��qR (1)

��0�qR (1)

�. Using

��qR (1)

�= 1 and the derivatives presented earlier, this amounts to showing:

e! �qR (1)�hp(1)�p(0)p(e!)�q

i e! + p (e!) [p(0)�q][p(e!)�q]2

<1

�1 + q [p(0)�q][p(1)�q]2

A su¢ cient condition for the preceding to be true is q

[p(1)�q]2 <p(e!)

[p(e!)�q]2 , which is guaranteed


by p (1) > p (e!) > q. Therefore, for � = 0, Y (r) is smoother than YS (r). Since � = 0

puts all the weight on Yk�3 (r) in the calculation of Y (r), we can conclude that Yk�3 (r) is

smoother than YS (r). As � increases, weight shifts from Yk�3 (r) to Y1 (r) so, to show that

Y (r) is smoother than YS (r) for all �, it will be enough to show that Y (r) is smoother than

YS (r) for � = 1. In this case, what we want to establish is:

��qR (1)

��0�qR (1)

�+ e! �qR (1)� e!0 �qR (1)� < 2� �qR (1)� �0 �qR (1)�

Substituting in the derivatives (and suppressing the argument r), the above inequality be-

comes:

� (1 + � [1� e!])[p (1)� p (0)] �1

�� +

hq� +

R 1�p (x) dx

i[p(0)�q][p(�)�q]2

� (E.1)

+e!

p0 (e!)R (e!) + p (e!)R0 (e!) < 2

[p (1)� p (0)] �1

��1 + q [p(0)�q]

[p(1)�q]2

�A su¢ cient condition can be established by noting the following:

� (1 + � [1� e!])[p (1)� p (0)] �1

�� +

hq� +

R 1�p (x) dx

i[p(0)�q][p(�)�q]2

� < 2� e![p (1)� p (0)] �1

��1 + q [p(0)�q]

[p(1)�q]2

�In particular, substituting this upper bound into (E.1) and rearranging yields the su¢ cient

condition below:

[p (1)� p (0)] �1

��1 + q [p (0)� q]

[p (1)� q]2

�< p0 (e!)R (e!) + p (e!)R0 (e!)

Replacing R (e!) and R0 (e!) with the appropriate expressions, the su¢ cient condition thenbecomes:


q[p (0)� q]

[p (1)� q]2< 1 +

[p (e!)� p (0)]

p (e!)� q+ p (e!) [p (0)� q]

[p (e!)� q]2

This inequality is certainly satis�ed if q

[p(1)�q]2 < p(e!)[p(e!)�q]2 , which is again guaranteed by

p (1) > p (e!) > q. Therefore, Y (r) is smoother than YS (r) when smoothness is de�ned

according to S.

Consider now bS. For a decreasing function y (r), we can write:bS = �Z b

a

y0 (r) dr = � [y (b)� y (a)]

Since Y (0) = YS (0), establishing that Y (r) is smoother than YS (r) when smoothness

is de�ned according to bS amounts to establishing Y �qR (1)� > YS�qR (1)

�. A su¢ cient

condition is max�e! �qR (1)� ; � �qR (1)� < �

�qR (1)

�= 1, which was shown to be true in

Section D. �

F. Proof of Proposition 5

Suppose an unexpected shock hits at t = 1 (i.e., "1 6= 0 but "t = 0 for all t > 1) and an

additional T � 1 periods are required for the policy rate to return to its steady state value.

Without relationship lending, output at time t is just YS (rt), where YS (�) is as de�ned in

(14). The variance of the transition path relative to steady state in the standard credit

model is thus:

�2S �1

T

TXt=1

[YS (rt)� YS (rss)]2

The analogous measure for the model with relationship lending is:

�2R �1

T

TXt=1

[Yt (rt)� Y (rss)]2

Note that Yt (rt) is not necessarily equal to Y (rt). In addition to the current cost of funds,


�rst period output depends on expectations of future relationship pro�ts and, therefore,

expectations of future policy rates. With rt+1 = rss + � (rss � rt) for t � 1, the appropriate

�rst period cuto¤ at time t is:

�t � arg min!2[0;1]

��hq! + R 1! p (x) dxiR (!) + �R 1e!(rt+1)J2 (rt+1jx) dx� rt

��Expected relationship pro�ts equal

R 1e!(r)J2 (rjx) dx and, at high policy rates, this expressionis decreasing in r. If "1 > 0, then rt+1 < rt for all t � 1 and, therefore,

R 1e!(rt+1)J2 (rt+1jx) dx >R 1e!(rt)J2 (rtjx) dx. We know from Lemma 2 that hq! + R 1! p (x) dxiR (!) is increasing in ! soit must be the case that �t < � (rt), where � (�) is as de�ned in (9). In other words, Yt (rt) >

Y (rt). Since "1 > 0 yields policy rates that are always above rss along the transition path, we

can further conclude that Y (rss) > Yt (rt) > Y (rt) and thus �2R <1T

PTt=1 [Y (rt)� Y (rss)]

2.

If instead "1 < 0, a similar argument establishes Y (rss) < Yt (rt) < Y (rt) and, once again,

�2R <1T

PTt=1 [Y (rt)� Y (rss)]

2.

A su¢ cient condition for �2R < �2S is thenPT

t=1 [Y (rt)� Y (rss)]2 <

PTt=1 [YS (rt)� YS (rss)]

2

and, since rt 2 [min frss; r1g ;max frss; r1g] � r along the transition path, having jY 0 (�)j <

jY 0S (�)j over this interval would guarantee it. Recalling the expressions for Y 0 (r) and Y 0

S (r)

from the proof of Proposition 4, what we would like to show is:

�1+�

� (r) �0 (r) + 11+�e! (r) e!0 (r) < � (r) �0 (r)

Consider �rst � = 0. Since e! (r) < � (r) for large values of r, it will be enough to show

that e!0 (r) < �0 (r) or, equivalently:

�� +hq� +

R 1�p (x) dx

i [p (0)� q]

[p (�)� q]2<

�p (1)� p (0)

p (e!)� q

� e! + p (e!) [p (0)� q]

[p (e!)� q]2

A su¢ cient condition for the above inequality is q� +R 1�p (x) dx < p (e!). To see that this

is indeed true, recallhq� +

R 1�p (x) dx

iR (�) = r = p (e!)R (e!) and R0 (�) > 0. With e! < �,


we have R (e!) < R (�) and, thus, q� +R 1�p (x) dx < p (e!). We can now conclude that

jY 0 (r)j < jY 0S (r)j for r 2 r when � = 0.

Note that � = 0 puts all the weight on Yk�3 (r) so what we have just shown is that

Yk�3 (r) is �atter than YS (r) for r 2 r. As � increases, weight shifts from Yk�3 (r) to Y1 (r).

Since Yk�3 (r) is assigned the smallest weight when � = 1, it will be enough to show that

jY 0 (r)j < jY 0S (r)j for r 2 r under � = 1 in order to show that jY 0 (r)j < jY 0

S (r)j for r 2 r

under any �.

Consider then � = 1. What we want to show now is � (r) �0 (r)+e! (r) e!0 (r) < 2� (r) �0 (r).To do so, de�ne h (!) � �! +

hq! +

R 1!p (x) dx

i[p(0)�q][p(!)�q]2 and rewrite �

0 (r) and �0 (r) as:

�0 (r) =1

[p (1)� p (0)] �1h (�)

�0 (r) =1 + � [1� e!]� �

R 1b! p(x)q dx

[p (1)� p (0)] �1h (�)<

2� e![p (1)� p (0)] �1h (�)

Since h0 (�) < 0 and � < �, we know that h (�) > h (�) and, therefore, �0 (r) < 2�e![p(1)�p(0)]�1h(�) .

A su¢ cient condition for what we currently want to show is thus e!0 (r) < �[p(1)�p(0)]�1h(�) .

Substituting in for e!0 (r), this su¢ cient condition becomes:1

p(e!)�p(0)p(e!)�q + p (e!) [p(0)�q]

[p(e!)�q]2<

�

�� +hq� +

R 1�p (x) dx

i[p(0)�q][p(�)�q]2

(F.1)

Recall that p (e!)R (e!) = hq� + R 1�p (x) dx

iR (�) and, under p (�) linear, R (!) = [p(1)�p(0)]�1!

p(!)�q .

Combining these equations yields q� +R 1�p (x) dx = p (e!) e![p(�)�q]

�[p(e!)�q] and lets us write (F.1) as:

�1 + p (e!)� p (0)� q

p (e!)� q

�� e!�2

1

[p (�)� q]� 1

[p (e!)� q]

�<p (e!)� p (0)

p (e!)� q

With 0 < e! < �, a su¢ cient condition for the preceding inequality is p (e!) h 1�� 1i<

2p (e!)� p (0)� q or, equivalently, � > p(e!)3p(e!)�p(0)�q . In other words, (F.1) is guaranteed for �

su¢ ciently large. Since �0 (r) > 0, it then follows that (F.1) is guaranteed for r su¢ ciently

large. �


Supplementary Appendix:An Extension with Distributional Dynamics

The results of the main text suggest that, on average, the informational properties of rela-

tionship lending lead to improved credit terms. In reality though, borrowers with improved

terms may be better able to overcome adverse idiosyncratic events that would have oth-

erwise put them out of business. Along with learning then, relationship lending may also

foster lower �rm exit rates. To understand the implications of this possibility, I extend the

baseline model. In particular, as long as a borrower stays with his insider, he experiences

exogenous separation with probability � � ", where " > 0. If or once he switches to an

outsider, separation occurs with probability �.

As noted in Subsection 6.2, the di¤erence in separation rates has two important impli-

cations. First, it gives insiders more bargaining power over high types so insiders can now

charge slightly above the outsider o¤er without losing these borrowers. Second, it provides

a new source of transition dynamics by making the policy rate a¤ect the distribution of

borrowers across periods. In this appendix, I set up the extended model more formally.

Value Functions

Outsiders still compete against each other and make zero expected pro�ts so their k � 3and k = 2 value functions are of the same form as those in Subsections 3.1 and 3.2. In

contrast, the k = 2 value function of an insider is now:

J2;I (rj!; d) = max(0;

(maxR

(Rj!)R� r + � (1� �+ ") Jk�3;I (rj!)

s:t: V2;I (!jR) � V2;U (!jR2;U;d)

))(S.1)

where V2;I (!jR) is the value of a second-time type ! borrower who stays with his insiderand pays loan rate R while V2;U (!jR2;U;d) is the value of this borrower should he move to anoutsider charging R2;U;d. The insider�s value function for any k � 3 is also given by the righthand side of (S.1) but with Vk�3;I (!jR) � Vk�3;U (!jRk�3;U;!) as the borrower�s participationconstraint.

For the baseline model (i.e., " = 0), it was proven that second-time borrowers are not

divided according to default history. In the absence of an analytical solution for the extended

model, there is no presumption that this is still the case. Therefore, second period loan rates

are not restricted to be history-independent and the �rst period borrower strategy is now

denoted by 1 (Rj!) to distinguish it from (Rj!). The value function of a �rst period lenderis then similar to equation (8) in the main text except that 1 (�) is used instead of (�) andexpected future pro�ts are determined using (S.1) and the market splitting that results.


Consider now the borrower side. For a �rst-time borrower facing loan rateR, the expected

payo¤s associated with choosing P1 and P2 are given by (S.2) and (S.3) respectively:

p (!) [�1 �R + �V2 (!jR2;I;N ; R2;U;N)] + (1� p (!)) �V2 (!jR2;I;D; R2;U;D) (S.2)

q [�2 �R + �V2 (!jR2;I;N ; R2;U;N)] + (1� q) �V2 (!jR2;I;D; R2;U;D) (S.3)

where V2 (!jR2;I;d; R2;U;d) = max fV2;I (!jR2;I;d) ; V2;U (!jR2;U;d)g is the borrower�s secondperiod value function. His �rst period value, V1 (!jR), is given by the maximum of (S.2) and(S.3) and his strategy is 1 (Rj!) = p (!) if and only if (S.2) is greater than (S.3). Finally,

to determine V2 (!jR2;I;d; R2;U;d), note that:

V2;I (!jR) = Vk�3;I (!jR) = max fp (!) [�1 �R] ; q [�2 �R]g+ � (�� ")R 10V1 (xjR1) dF (x)

+� (1� �+ ")max fVk�3;I (!jRk�3;I;!) ; Vk�3;U (!jRk�3;U;!)g

V2;U (!jR) = Vk�3;U (!jR) = max fp (!) [�1 �R] ; q [�2 �R]g+ ��R 10V1 (xjR1) dF (x)

+� (1� �)Vk�3;U (!jRk�3;U;!)

Transition Dynamics

Suppose an unanticipated permanent increase in r occurs at date t. Lenders with ad-

vanced borrowers can adjust immediately to the new steady state but this may not be true

for lenders with intermediate borrowers. Recall that the only piece of information available

to a second period outsider is whether the borrower defaulted on his �rst period loan and,

at date t, this outcome depends on loan rates induced by the t � 1 policy rate. In the in-sider�s problem, however, expected future pro�ts depend on loan rates induced by the t+ 1

policy rate. Since an equilibrium is reached when each lender�s o¤er is a best response to the

other�s, the second period loan rates that prevail at date t depend on both pre-shock and

post-shock policy rates. In contrast, the post-shock steady state is conditioned entirely on

the post-shock policy rate so the k = 2 equilibrium at date t may di¤er from the new k = 2

steady state.

To determine how long it takes to reach the new steady state, consider the market for

new borrowers. If �rst period lenders at date t expect a full adjustment by date t+ 1, then

they will adjust immediately. As a result, both outsider information and insider pro�ts at

date t+1 will be conditioned on the new policy rate. This means that the k = 2 equilibrium

will reach the new steady state by date t+1, consistent with the time t expectations of �rst

period lenders. In what follows, I focus on this case. That is, all contracts adjust to the new

steady state by date t + 1.19 Note, however, that even with a quick contract response, the19Other assumptions about the time it takes for contracts to adjust would be ad hoc at this point. Issues

of contract stickiness are thus left for future work.


e¤ects of the policy rate shock continue to be propagated through the distribution. We can

see this by formalizing the borrower �ows. Let I2;d;t (!) be an indicator function that equals

1 if a second-time borrower with type ! and default history d stays with his insider at date t.

Similarly, de�ne Ik�3;t (!) so that Ik�3;t (!) = 1 if Vk�3;I (!jRk�3;I;!;t) � Vk�3;U (!jRk�3;U;!;t).The mass of k = 1 borrowers at date t+ 1 is now:

1;t+1 = (�� ")R 10

� 2;I;t (x) + k�3;I;t (x)

�dx+ �

R 10

� 2;U;t (x) + k�3;U;t (x)

�dx

where 2;I;t (�) is the distribution of borrower types across k = 2 insiders and 2;U;t (�) is thedistribution of borrower types across k = 2 outsiders. The corresponding distributions for

k � 3 are denoted by k�3;I;t (�) and k�3;U;t (�). The laws of motion for these distributionsare as follows:

2;I;t+1 (!) = 1;t

" 1 (R1;tj!) I2;N;t+1 (!)+ [1� 1 (R1;tj!)] I2;D;t+1 (!)

#

2;U;t+1 (!) = 1;t

" 1 (R1;tj!) [1� I2;N;t+1 (!)]

+ [1� 1 (R1;tj!)] [1� I2;D;t+1 (!)]

#

k�3;I;t+1 (!) = (1� �+ ")� 2;I;t (!) + k�3;I;t (!)

�Ik�3;t+1 (!)

k�3;U;t+1 (!) =

"(1� �)

� 2;U;t (!) + k�3;U;t (!)

�+(1� �+ ")

� 2;I;t (!) + k�3;I;t (!)

�[1� Ik�3;t+1 (!)]

#

Shocks to the policy rate a¤ect the terms o¤ered by various lenders and changes in

these terms then a¤ect which borrowers choose to stay with their insiders (i.e., I2;N , I2;D,

and Ik�3 respond). When " is positive, types that stay with their insiders become more

persistent so changes in r alter the distribution of borrower types in and across periods. As

these distributions evolve to their new steady states, aggregate dynamics are observed well

beyond time t.

Numerical Analysis

To compute the equilibrium quantities, discretize the type space and the set of possible

loan rates and initialize the loan rate functions and the value functions. Given the loan

rates, I determine the borrowers�strategies by iterating on their value functions then, based

on these strategies, I iterate on the loan rates to �nd the optimal lender responses. The

equilibrium is determined by iterating on the outer loop until the starting and ending loan

rate functions converge.


To execute the iterations, I set � = 0:96 and use a uniform distribution of types. Returns

are �1 = 5 and �2 = 6 so that the speculative project yields 20 percent more than the invest-

ment project if successful. The probability of success for the investment project is assumed

to be linear in borrower type, satisfying p (!) = p (0) + [p (1)� p (0)]!. By de�nition, the

best type is very likely to succeed if he operates P1 so I set p(1) = 0:9. The success rate

of the speculative project is much lower but, since it still needs to be a legitimate outside

option, I consider q = 0:65. With values for q, �1, and �2, we can then use p(0)�1 = q�2 to

pin down p (0) = 0:78. Unless otherwise speci�ed, � = 0:3 and " = 0:065.

Steady State

Figure S.1 illustrates steady state results for the extended model. The aggregate output

pro�le resembles that in the main text but, by making better types more persistent, higher

values of " shift the pro�le upwards. There are two additional di¤erences relative to the

baseline model. First and as shown in Figure S.1(a), the steady state measure of rela-

tionship lending exhibits a hump-shaped response to increases in the policy rate instead of

a monotonic decline. Second and as shown in Figure S.1(b), credit history matters even

though the insider still discovers his borrower�s type with certainty (i.e., � = 1).

A higher policy rate increases the cost of lending so, all else constant, the lowest type

on which the insider breaks even rises. As before then, insiders become more selective in

their retention of borrowers and fewer lending relationships are formed. Now, however, the

additional bargaining power that " gives the insider over better types means that more of

the necessary break even can be accommodated by increases in the loan rate, stemming the

restriction of insider credit. The bargaining power e¤ect plays out initially but is eventually

dominated by the selectivity e¤ect, leading to the hump-shaped response in relationship

lending.

The bargaining and selectivity e¤ects are also useful for understanding why credit history

can matter with " > 0. At higher policy rates, the increase in insider selectivity means that

more types have to resort to outsider credit. This increases outsider uncertainty and makes

credit history a natural screening mechanism. The informativeness of credit history, however,

depends on the �rst period loan rate. In particular, a very high R1 induces most types to

choose P2 in the �rst period and implies high default probabilities across the board. The

opposite is true when R1 is very low. Therefore, by getting good �rms to choose P1 and bad

�rms to choose P2, moderate �rst period loan rates generate the most informative credit

histories. For credit history to matter then, we need a relatively high value of r but a

relatively moderate value of R1. This con�guration can be achieved under " > 0 since the

bargaining power a¤orded to insiders over high types increases the expectation of future


pro�ts and, for any r, competitive �rst period lenders settle on an even lower value of R1.

To be sure, adverse selection still arises with " > 0 but, for relatively high policy rates, the

informativeness of credit history is such that d cannot be completely crowded out.

Dynamics

Consider now the transition between steady states after an increase in the policy rate at

date t. As shown in Figure S.2, the initial response of relationship lending tends to over-

shoot its new steady state value. To see why, de�ne the extensive margin in period k as

the total number of borrowers in that period and the intensive margin as the proportion of

these borrowers that enter into multi-period lending relationships. The extent of relation-

ship lending in any given period is approximately equal to the product of its intensive and

extensive margins.20 The extent of relationship lending at any given date is then equal to

the sum of the extents for periods 2 and above. The right panel of Figure S.1(c) reveals

that k = 2 is critical for the analysis. When r increases from 0:5 to 0:75, the bargaining

power e¤ect drives up the second period intensive margin and we observe the immediate

increase in relationship lending shown in Figure S.2(a). Over time though, more lending

relationships mean fewer exogenous separations so the distribution of borrowers eventually

shifts away from k = 2 and the extent of relationship lending declines along the transition

path. Therefore, the increase in relationship lending overshoots its new steady state and the

decrease in output undershoots. As set up in Subsection 4.5, the standard model adjusts

to its new steady state immediately so the results presented here suggest that relationship

lending leads to a more gradual transition after certain permanent shocks.21

In comparison, Figure S.2(b) demonstrates that an increase in the policy rate from 0:75

to 1 causes the decrease in output to overshoot. For this range of r, the insider�s selectivity

e¤ect dominates, pushing the second period intensive margin back down. As the immediate

decrease in relationship lending eventually increases the number of young borrowers (i.e.,

the pool of potential relationship borrowers), the initial declines in relationship lending and

output are partially o¤set over time. Even at its trough, however, aggregate output in Figure

S.2(b) exceeds the standard credit model�s new steady state of YS (1) � 4:16.

20The result is an approximation for the second period since it aggregates across default histories.21This is not to say that traditional models do not generate dynamics. Instead, the transitions presented

here should be interpreted as dynamics over and above those generated by a model that ignores relationshiplending.


Figure S.1: Steady state results in the extended model

(a) Aggregate output and relationship lending

(b) Realized second period loan rates

(c) Intensive and extensive margins


Figure S.2: Dynamics in the extended model

(a) Transition between steady states, r = 0:5 to r = 0:75

(b) Transition between steady states, r = 0:75 to r = 1

relationship lending and the transmission of monetary...

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